this article describes how to deploy dihedral group theory to detect Facial Key Points (FKP) symmetry to recognize emotions. The method can be applied in many other areas which those have the same data texture. Key-Words: Emotion analysis, Facial expression, Facial key points, Geometric features, Dihedral group, Symmetry, Rotation, Reflection.

thirty nine novel 1,2,3,4-tetrahydropyrimidinone (thione)s were subjected to conformational studies. density functional theory at b3lyp/6-31 g* was performed as the computational method of high accuracy. important dihedral angles and bond lengths were investigated and the values obtained were explainable. results of this work confirm a twisted boat tetrahydropyrimidine ring conformation with an...

The title compound, C(18)H(18)FNO(3), consists of three individually planar subunits, namely two substituted benzene rings and one amino-acrylate group. The dihedral angle between the two benzene rings is 47.48 (8)°. The amino-acrylate group forms dihedral angles of 57.95 (7) and 11.27 (6)° with the methoxy-phenyl and fluorophenyl rings, respectively.

The title compound, C(20)H(16)N(2), has been synthesized by the reaction of benzaldehyde with o-phenyl-endiamine and l-proline. The benzimidazole group makes a dihedral angle of 29.04 (1)° with the attached benzene ring, and is approximately perpendicular to the plane of the benzyl group [dihedral angle = 88.9 (1)°] The crystal packing exhibits no unusually short inter-molecular contacts.

We describe a simple locally CAT(0) classifying space for XXL type Artin groups (with all labels at least 5). Furthermore, when the group is not dihedral, we rank 1 periodic geodesic, thus proving that are acylindrically hyperbolic. Together with Property RD proved by Ciobanu, Holt and Rees, property implies Baum–Connes conjecture groups.

In the title compound, C17H27N3OS2, the amide group is approximately coplanar with the pyridine ring [dihedral angle = 1.6 (1)°], whereas the di-thio-carbamate group is nearly perpendicular to the pyridine ring [dihedral angle = 76.7 (1)°]. In the crystal, pairs of weak C-H⋯O hydrogen bonds link the mol-ecules into inversion dimers.

The study of entanglement properties multi-qubit states that are invariant under permutations qubits is motivated by potential applications in quantum computing, communication, and metrology. In this work, we generalize the notions symmetrization, Dicke states, Majorana representation to alternating, cyclic, dihedral subgroups full group permutations. We use these tools characterize analyze the...

We construct a pair of non-commutative rank 8 association schemes from 3 non-symmetric scheme. For the pair, two have same character table but different Frobenius-Schur indicators. This situation is similar to dihedral group and quaternion order 8. also determine structures adjacency algebras them over rational number field.